UCT EEE2047S / 2021
A signal, harmonic by harmonic
Completed symbolic/numerical Fourier tasks, with a new interactive reconstruction derived from the submitted period-eight pulse.
UCT / COURSE CONTEXT / 2021
EEE2047S · Signals and Systems I
Linear systems, convolution, Fourier analysis and sampling in time and frequency domains.
2026 PORTFOLIO RECREATIONS
Harmonic by harmonicEEE2047S / FOURIER RECONSTRUCTION
Odd harmonics through N = 1
x_N(t) = 1/2 + Σ [2/(kπ)] sin(kπt/4), k oddNew browser visualisation of the period 8, 0≤t<4 pulse in completed Lab 2 Question 2. Dashed: ideal pulse; faint: individual harmonics with DC; solid: finite sum. Not a hardware measurement. Ringing near discontinuities remains.
ENGINEERING PERSPECTIVE
Inside the work
Show how a periodic signal can be represented by harmonics. The pulse makes finite approximation, edge sharpness and ringing visible as the number of terms changes.
- SymPy
- Symbolic integration of complex Fourier coefficients in the submitted tasks.
- NumPy / Python
- Evaluate finite sums and generate spectral/reconstruction plots.
Define the period
Use the submitted pulse: one over the first half of an eight-unit period, zero over the second.
- Fourier coefficients
- The period-eight half-duty pulse has a DC term of 1/2 and odd sine terms 2/(kπ) sin(kπt/4).
- Finite approximation
- Adding harmonics improves the shape away from discontinuities but does not eliminate edge ringing.
- Phase conventions
- A shifted pulse changes coefficient phase; frequency and time definitions must remain consistent.
What the demonstration establishes
The browser plot is a new mathematical explanation derived from the submitted function. It is not a measured waveform or an original application screenshot.
Evidence behind this page
Named Lab2 submission with completed task calculations and retained spectral figures.
How it works
Integrate the period-eight pulse x(t) = 1 for 0 ≤ t < 4 to obtain complex Fourier coefficients. The real reconstruction has a DC term of 1/2 and odd sine terms 2/(kπ) sin(kπt/4). The submitted tasks also analyse triangular and ramp-like signals.
Results & recovery
- Named submission retains actual spectral plots and completed SymPy integration.
- The new finite-sum visual shows how harmonic count changes edge sharpness and ringing. It is a mathematical explanation, not an original lab screenshot.
Technical limitations
No new physical signal measurement. The finite sum does not remove discontinuity ringing; worksheet and personally completed task outputs must not be conflated.
What I would improve now
Add numerical coefficient checks, convergence/error plots away from discontinuities and explicit phase-convention tests.